**Question.**
Let $C_1,\dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly,
each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles
have the length prescribed by the partition.) Give a necessary and sufficient condition on
$C_i$ that would ensure that there are permutations $\sigma_i\in C_i$ with
$$\prod\sigma_i=1.$$

**Variant.**
Same question, but now $\sigma_i$'s are required to be *irreducible* in the sense that
they have no common invariant proper subsets $S\subset\lbrace 1,\dots,n\rbrace$.

I am not certain how hard this question is, and I would appreciate any comments or observations. (I was unable to find references, but perhaps I wasn't looking for the right things.)

This question is inspired by Jonah Sinick's question via the simple

**Geometric interpretation.**
Consider the Riemann sphere with $k$-punctures $X=\mathbb{CP}^1-\lbrace x_1,\dots,x_k\rbrace$.
Its fundamental group $\pi_1(X)$ is generated by loops $\gamma_i$ ($i=1,\dots,k$) subject to the relation
$$\prod\gamma_i=1.$$
Thus, homomorphisms $\pi_1(X)\to S_n$ describe degree $n$ covers of $X$, and the problem
can be stated as follows: Determine whether there exists a cover of $X$ with prescribed
ramification over each $x_i$. The variant requires in addition the cover to be irreducible.

**Background.**
The Deligne-Simpson Problem refers to the following question:

Fix conjugacy classes $C_1,\dots,C_k\in\mathrm{GL}(n,\mathbb{C})$ (given explicitly by $k$ Jordan forms). What is the necessary and sufficient condition for existence of matrices $A_i\in C_i$ with $$\prod A_i=1$$ (variant: require that $A_i$'s have no common proper invariant subspaces)?

There are quite a few papers on the subject; my favorite is Simpson's paper, which has references to other relevant papers. The problem has a very non-trivial solution (even stating the answer is not easy): first there is a certain descent procedure (*Katz's middle convolution algorithm*) and then the answer is constructed directly (as far as I understand, there are two answers: Crawley-Boevey's argument with parabolic bundles, and Simpson's construction using non-abelian Hodge theory).

The same geometric interpretation shows that the usual Deligne-Simpson problem is about finding local systems (variant: irreducible local systems) on $X$ with prescribed local monodromy.

So: any remarks on what happens if we go from $\mathrm{GL}(n)$ to $S_n$?